Singular measure

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In mathematics, two positive (or signed or complex) measures μ and ν defined on a measurable space (Ω, Σ) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. This is denoted by

As a particular case, a measure defined on the Euclidean space Rⁿ is called singular, if it is singular in respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure.

[edit] See also

• Lebesgue's decomposition theorem
• Absolutely continuous
• Singular distribution

[edit] References

• Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. ISBN 1-58488-347-2.
• J Taylor, An Introduction to Measure and Probability, Springer, 1996. ISBN 0-387-94830-9.

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This article incorporates material from singular measure on PlanetMath, which is licensed under the GFDL.